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On the automorphism group of a binary q-analog of the Fano plane Michael Braun 5 1 Faculty of Computer Science, University of Applied Sciences, Darmstadt, Germany 0 2 Michael Kiermaier l u Lehrstuhl II fu¨r Mathematik, Universita¨t Bayreuth, Germany J Anamari Nakić 6 1 Faculty of Electrical Engineering and Computing, Universityof Zagreb, Croatia ] O C . h Abstract t a m The smallest set of admissible parameters of a q-analog of a Steiner system is [ S2[2,3,7]. The existence ofsuch a Steiner system– knownasa binary q-analog of the Fano plane – is still open. In this article, the automorphism group of a 2 putative binary q-analog of the Fano plane is investigated by a combination of v theoreticalandcomputationalmethods. As a conclusion,it is either rigidorits 0 9 automorphism group is cyclic of order 2, 3 or 4. Up to conjugacy in GL(7,2), 7 there remains a single possible group of order 2 and 4, respectively, and two 7 possible groups of order 3. For the automorphisms of order 2, we give a more 0 general result which is valid for any binary q-Steiner triple system. . 1 0 5 1. Introduction 1 : v Duetotheapplicationinerror-correctioninrandomizednetworkcoding[20], i X q-analogsofcombinatorialdesignshavegainedalotofinterestlately. Arguably the most important open problem in this field is the question of the existence r a of a 2-(7,3,1) design, as it has the smallest admissible parameterset of a non- q trivial q-Steiner system with t≥2. It is known as a q-analogof the Fano plane and has been tackled in many articles [11, 12, 16, 23, 24, 29, 30]. In this paper we investigate the automorphism group of a putative binary q-analog of the Fano plane. The following result will be proven in Section 4. Theorem 1. A binary q-analog of the Fano plane is either rigid or its automorphism group is cyclic of order 2, 3 or 4. Representing the automorphism group as a subgroup of GL(7,2), up to conjugacy it is contained in the following list: (a) The trivial group. 1 (b) The group of order 2 0100000 1000000 0001000 0010000 . * 0000010 + 0000100 0000001   (c) One of the following two groups of order 3: 0100000 0100000 1100000 1100000 0001000 0001000 0011000 and 0011000 . * 0000010 + * 0000100 + 0000110 0000010 0000001 0000001     (d) The cyclic group of order 4 1100000 0110000 0010000 0001100 . * 0000110 + 0000011 0000001   Theideaofeliminatingautomorphismgroupshasbeenusedintheexistence problems for other notorious combinatorial objects. Probably the most promi- nent example is the projective plane of order 10. In [31, 32] it was shown that the orderofanautomorphismis either 3 or5. Those twoordershadbeenelim- inated later in [1, 13], implying that the automorphism group of a projective plane of order 10 must be trivial. Finally, the non-existence has been shown in [22] in an extensive computer search. There are other examples of existence problems where the same idea has been used, that remain still open. In coding theory, the question of the existence of a binary doubly-even self-dual code with the parameters [72,36,16] wasraisedmorethan40yearsagoin[27]. Itsautomorphismshavebeenheavily investigated in a series of articles. After the latest result [33], it is known that its automorphism group is either trivial, cyclic of order 2, 3 or 5, or a Klein four group. In classical design theory, the smallest v for which the existence of a 3-design of order v is undecided is 16; indeed a 3-(16,7,5) design is still unknown. Recentresultsshowthatifsuchdesignexists,thenitsautomorphism group is either trivial or a 2-group [10, 25]. The proof of Theorem 1 is partially based on the following more general result on automorphisms of order 2 of binary q-Steiner triple systems, which will be shown in Section 3. Theorem 2. Let D be a binary S2[2,3,v] q-Steiner triple system and A ∈ GL(v,2) the matrix representation of an automorphism of D of order 2. For s ∈ {1,...,⌊v/2⌋} let A denote the v×v block diagonal matrix built from s v,s blocks (01) followed by a (v−2s)×(v−2s) unit matrix. 10 (a) In the case v ≡1mod6, A is conjugate to a matrix A with 3|s. v,s (b) In the case v ≡3mod6, A is conjugate to a matrix A with s6≡2mod3. v,s 2 2. Preliminaries 2.1. Group actions Suppose that a finite group G is acting on a finite set X. For x ∈ X and α∈G, the image of x under α is denoted by xα. The set xG ={xα :α∈G} is called the orbit of x under G or the G-orbit of x, in short. The set of all orbits is a partition of X. We say that x∈X is fixed by G if xG ={x}. So the fixed elements correspond to the orbits of length 1. The set G ={α∈G : xα =x} x is a subgroup of G, called the stabilizer of x in G. For all g ∈G we have G =g−1G g. (1) xg x Inparticularthestabilizersofelementsinthesameorbitareconjugate,andany conjugate subgroup H of G is the stabilizer of some element in the orbit of x. x TheOrbit-StabilizerTheoremstatesthat#xG =[G:G ],1 implying that#xG x divides #G. The action of G on X extends in the obvious way to an action on the set of subsets of X: For S ⊆ X, we set Sα = {xα : x ∈ S}. For further reading on the theory of finite group actions, see [18]. 2.2. Linear and semilinear maps Let V,W be vector spaces over a field F. A map f : V → W is called semilinear ifitisadditiveandifthereisafieldautomorphismσ ∈Aut(F)such that f(λv) = σ(λ)f(v) for all λ ∈ F× and all v ∈ V. The map f is linear if and only if σ = id . In fact, the general linear group GL(V) of all linear F bijections of V is a normal subgroup of the general semilinear group ΓL(V) of all semilinear bijections of V. Moreover, ΓL(V) decomposes as a semidirect productGL(V)⋊Aut(F). Inthis representation,the naturalactiononV takes the form (φ,σ)(v) =φ(σ(v)), where σ(v) is the component-wise application of σ to the coordinate vector of v with respect to some fixed basis. The center Z(GL(V)) of GL(V) consists of all diagonal maps v 7→λv with × λ∈F . It is a normal subgroup of GL(V) and of ΓL(V). The quotient group GL(V)/Z(GL(V)) is known as the projective linear group PGL(V), and the quotient group ΓL(V)/Z(GL(V)) is known as the projective semilinear group PΓL(V). 2.3. The subspace lattice Let V denote a v-dimensional vector space over the finite field F with q q elements. For simplicity, a subspace of V of dimension r will be called an r- subspace. The set of all r-subspaces of V is called the Graßmannian and is denoted by V . As in projective geometry, the 1-subspaces of V are called r q points, the 2-subspaces lines and the 3-subspaces planes. Our focus lies on the (cid:2) (cid:3) caseq =2,wherethe1-subspaceshxiF2 ∈ V1 2 areinone-to-onecorrespondence (cid:2) (cid:3) 1Thesymbol#denotes thecardinalityofaset. 3 with the nonzero vectors x ∈ V \{0}. The number of all r-subspaces of V is given by V v (qv −1)···(qv−r+1−1) # = = . r r (qr−1)···(q−1) (cid:20) (cid:21)q (cid:20) (cid:21)q The set L(V) of all subspaces of V forms the subspace lattice of V. By the fundamental theoremof projectivegeometry,for v 6=2 the automorphism groupof L(V) is givenby the naturalaction of PΓL(V) onL(V). In the case that q is prime, the groupPΓL(V) reduces to PGL(V), and for the case of our interest q =2, it reduces further to GL(V). After a choice of a basis of V, its elements are represented by the invertible v×v matrices A, and the action on L(V) is given by the vector-matrix-multiplicationv7→vA. 2.4. Designs Definition 2.1. Let t,v,k be integers with 0 ≤ t ≤ k ≤ v, λ another positive integer and P a set of size v. A t-(v,k,λ) (combinatorial) design D is a set of k-subsets of P (called blocks) such that every t-subset of P is contained in exactly λ blocks of D. When λ=1, D is called a Steinersystem and denoted by S(t,k,v). If additionally t=2 and k =3, D is called a Steiner triple system. For the definition of a q-analog of a combinatorialdesign, the subset lattice onP isreplacedbythesubspacelatticeofV. Inparticular,subsetsarereplaced by subspaces and cardinality is replaced by dimension. So we get [7, 8]: Definition 2.2. Let t,v,k be integers with 0 ≤ t ≤ k ≤ v, λ another positive integer and V an F -vector space of dimension v. A t-(v,k,λ) design D over q q thefieldF isasetofk-subspacesofV (called blocks)suchthateveryt-subspace q of V is contained in exactly λ blocks of D. When λ=1, D is called a q-Steiner system and denoted by S [t,k,v]. If additionally t=2 and k =3, D is called a q q-Steiner triple system. There are good reasons to consider the subset lattice as a subspace lattice overtheunary“field”F1[9]. Thus,classicalcombinatorialdesignscanbeseenas thelimitcaseq =1ofadesignoverafinitefield. Indeed,quiteafewstatements about combinatorial designs have a generalization to designs over finite fields, such that the case q =1 reproduces the original statement [6, 15, 16, 26]. Oneexampleofsuchastatementisthefollowing[28,Lemma4.1(1)]: IfD is a t-(v,k,λ ) design, then D is also an s-(v,k,λ ) for all s∈{0,...,t}, where t q s q v−s t−s q λ :=λ . s t k−s (cid:2) (cid:3) t−s q (cid:2) (cid:3) In particular, the number of blocks in D equals v t q #D =λ0 =λt k . (cid:2) (cid:3) t q (cid:2) (cid:3) 4 Soanecessaryconditionontheexistenceofadesignwithparameterst-(v,k,λ) q isthatforall s∈{0,...,t}the numbersλ v−s / k−s areintegers(integrality t−s q t−s q conditions). Inthis case,the parametersett-(v,k,λ) is calledadmissible. Itis (cid:2) (cid:3) (cid:2) q(cid:3) further called realizable if a t-(v,k,λ) design actually exists. q For designs over finite fields, the action of Aut(L(V)) ∼= PΓL(V) on L(V) providesanotionofisomorphism. TwodesignsinthesameambientspaceV are calledisomorphic iftheyarecontainedinthesameorbitofthisaction(extended tothepowersetofL(V)). Theautomorphism group Aut(D)ofadesignD isits stabilizer with respect to this group action. If Aut(D) is trivial, we will call D rigid. Furthermore, for G ≤ PΓL(V), D will be called G-invariant if it is fixed by all g ∈G orequivalently, if G≤Aut(D). Note that if D is G-invariant,then D is also H-invariant for all subgroups H ≤G. 2.5. Admissibility and realizability The question ofthe realizability of anadmissible parameter setis very hard to answer in general. In the special case t = 1, q-Steiner systems are called spreads. It is known that the spread parameters S [1,k,v] are realizable if and q only if they are admissible if and only if k divides v. However for t ≥ 2, the problem tantalized many researchers [2, 4, 11, 23, 29, 30]. Only recently, the first example of such a q-Steiner system was constructed [4]. It is a q-Steiner triple system with the parameters S2[2,3,13]. As a direct consequence of the integrality conditions, the parameter set ofa SteinertriplesystemS(2,3,v)oraq-SteinertriplesystemS [2,3,v]isadmissible q ifandonlyifv ≡1,3mod6andv ≥7. Intheordinarycaseq =1,theexistence question is completely answered by the result that a Steiner triple system is realizable if and only if it is admissible [17]. However in the q-analog case, our current knowledge is quite sparse. The only decided case is given by the above mentioned existence of an S2[2,3,13]. The smallestadmissible case of a q-Steiner triple systemis S [2,3,7],whose q existenceisopenforanyprimepowervalueofq. Itisknownasaq-analog of the Fano plane, since the unique Steiner triple system S(2,3,7) is the Fano plane. It is worth noting that there are cases of Steiner systems without a q-analog, as the famous large Witt design with parameters S(5,8,24) does not have a q-analog for any prime power q [15]. 2.6. The method of Kramer and Mesner The method of Kramer and Mesner [21] is a powerful tool for the computational construction of combinatorial designs. It has been successfully adopted and used for the construction of designs over a finite field [24, 5]. For example, the hitherto only known q-analogof a Steiner triple systemin [4] has been con- structedby this method. Here we givea shortoutline, for more details we refer the reader to [5]. As another computational construction method we mention the use of tactical decompositions [14], which has been adopted to the q-analog case in [26]. 5 The Kramer-Mesner matrix MG is defined to be the matrix whose rows t,k and columns are indexed by the G-orbits on the set V of t-subspaces and on t q theset V ofk-subspacesofV,respectively. TheentryofMG withrowindex k q (cid:2) (cid:3) t,k TG and column index KG is defined as #{K′ ∈ KG | T ⊆ K′}. Now there (cid:2) (cid:3) exists a G-invariant t-(v,k,λ) design if and only if there is a zero-one solution q vector x of the linear equation system MGx=λ1, (2) t,k where 1 denotes the all-one column vector. More precisely, if x is a zero-one solution vector of the system (2), a t-(v,k,λ) design is given by the union of q all orbits KG where the corresponding entry in x equals one. If x runs over all zero-one solutions, we get all G-invariant t-(v,k,λ) designs in this way. q 2.7. Normal forms for square matrices Let F be a field and V a vector space over F of finite dimension v. Two elements of GL(V) are conjugate if and only if their transformation matrices are conjugate in the matrix group GL(v,F) if and only if the transformation matrices are similar in the sense of linear algebra. If F is algebraically closed, representativesofthe matricesinFv×v uptosimilarityaregivenbythe Jordan normal forms. In the following, we discuss two common normal forms for the moregeneralcasethatF is notalgebraicallyclosed,the Frobenius normalform and the generalized Jordan normal form. For a monic polynomial f =a0+a1X +...+an−1Xn−1+Xn ∈F[X], the companion matrix A of f is the n×n matrix over F defined as f 0 1 0 ··· 0  ... ... ... ... ...  Af = ... ... ... 0 .    0 ··· ··· 0 1    −a0 −a1 −a2 ··· −an−1   It is known that for any square matrix A over F there are unique monic invariant factors f1 | ... | fs ∈ F[X] such that A is conjugate to a block diagonal matrix consisting of the blocks A ,...,A . This matrix is called the f1 fs Frobenius normal form or the rational normal form of A. The last invariant factor f equals the minimal polynomial of A and the product of all the invari- s ant factors equals the characteristic polynomial of A. The Frobenius normal form corresponds to a decomposition of V into a minimum number of A-cyclic subspaces. (A subspace U is called A-cyclic if there exists a vector v∈V such that U =hv,Av,A2v,...i .) F 6 Byfurtherdecomposingtheinvariantfactorsintopowersofirreduciblepoly- nomials,onearrivesatanalternativematrixnormalform: Forapositiveinteger m and a monic irreducible polynomial f, we define the Jordan block A U 0 ··· 0 f .. .. ..  0 Af . . .  Jf,m = ... ... ... ... 0     ... ... A U   f   0 ··· ··· 0 A   f   with m consecutive blocks A on the diagonal. Here, 0 denotes the n×n zero f matrixandU denotesthen×nmatrixwhoseonlynonzeroentryisanentry1in the lowerleft corner. Eachsquarematrix overF is conjugate to a (generalized) Jordan normal form, which is a block diagonal matrix consisting of Jordan blocks J ,...,J with monic irreducible polynomials g . Furthermore, g1,m1 gs,ms i the Jordan normal form is unique up to permuting the Jordan blocks. The JordannormalformcorrespondstoadecompositionofV intoamaximalnumber of A-invariant subspaces. For that reason, the matrix blocks of the Jordan normal form are typically smaller than those in the Frobenius normal form. Depending on the application, this might be an advantage. 3. Automorphisms of order 2 of binary q-Steiner triple systems In this sectionwe proveTheorem2, which is a resultonthe automorphisms of order 2 of a binary q-Steiner triple system. Lemma 3.1. In GL(v,2) there are exactly ⌊v/2⌋ conjugacy classes of elements of order 2. Representatives are given by the block-diagonal matrices A with v,s s ∈{1,...,⌊v/2⌋}, consisting of s consecutive 2×2 blocks (01), followed by a 10 (v−2s)×(v−2s) unit matrix. Proof. Let A ∈GL(v,2). The matrix A is of order 2 if and only if its minimal polynomialisX2−1=(X+1)2. Equivalently,allthe invariantfactorsofAare X+1 or(X+1)2, andthe invariantfactor (X+1)2 must appearat leastonce. So for the number s of the invariant factor (X +1)2, there are the possibilities s∈{1,...,⌊v/2⌋}. Representativesof the elements of order 2 are now givenby the associated Frobenius normal forms. The invariant factor X +1 gives 1×1 blocks (1), and the invariant factor (X+1)2 =X2+1 gives 2×2 blocks (01). 10 By putting the 1×1 blocks at the end rather than at the beginning, we attain the matrices A . v,s ForamatrixAoforder2,theuniqueconjugateA givenbyLemma3.1will v,s be calledthe type ofA. If Ais anautomorphismofadesignD, by equation(1) thereisanisomorphicdesignhavingtheautomorphismA . Therefore,forthe v,s investigation of the action of hAi on D, we may assume A = A without loss v,s of generality. 7 Lemma 3.2. Let A be a matrix of order 2 and type A . The action of hAi v,s partitions the point set Fv2 into 2v−s−1 fixed points and 2v−s−1(2s−1) orbits 1 q of length 2. (cid:2) (cid:3) Proof. MemayassumeA=A . BytheOrbit-StabilizerTheoremand#hA i= v,s v,s 2, all orbits are of length 1 or 2. From the explicit form of the matrices A , it v,s is clear that a vector x=(x1,x2,...,xv)∈Fv2 is fixed by Av,s if and only if x1 =x2, x3 =x4,..., x2s−1 =x2s. Thus,eachofthese2v−s−1vectorsformsanorbitoflength1,andtheremaining (2v−1)−(2v−s−1)=2×2v−s−1(2s−1)vectorsarepairedinto 2v−s−1(2s−1) orbits of length 2. Example 3.3. A model of the classical Fano plane is given by the points and the planes in PG(2,2) = PG(F3). Its automorphism group is GL(3,2). By 2 Lemma 3.1, there is a single conjugacy class of automorphism types of order 2, represented by the matrix 0 1 0 A3,1 = 1 0 0 .   0 0 1   F3 The action of hA3,1i partitions the point set 12 2 into the 3 fixed points (cid:2) (cid:3) h(0,0,1)iF , h(1,1,0)iF , h(1,1,1)iF , 2 2 2 and the two orbits of length 2 {h(1,0,0)iF , h(0,1,0)iF } and {h(1,0,1)iF , h(0,1,1)iF }. 2 2 2 2 Example 3.4. By Lemma 3.1, the conjugacy classes of elements of order 2 in GL(7,2) are represented by 0100000 0100000 0100000 1000000 1000000 1000000 0010000 0001000 0001000 A7,1 =0001000, A7,2 =0010000, A7,3 =0010000. 0000100 0000100 0000010 0000010 0000010 0000100 0000001 0000001 0000001       The numberoffixedpoints is 63,31and15,and thenumberoforbits oflength2 is 32, 48 and 56, respectively. Now we investigate the automorphisms of order 2 of an S2[2,3,v] q-Steiner triple system D. For the remainder of the article, we will assume that V =Fv. 2 This allows us to identify GL(V) with the matrix group GL(v,2). Lemma 3.5. Let D be an S2[2,3,v] q-Steiner system with an automorphism A of order 2 and type A . Then each block of D fixed under the action of hAi v,s 8 contains either 3 or 7 fixed points. The number of fixed blocks of the first type is F3 =2v−s−2(2s−1) (3) and the number of fixed blocks of the second type is 22v−2s−1+1−3·2v−s−2(2s+1) F7 = . (4) 21 Proof. Without restriction A = A . Let F denote the set of blocks of D v,s fixed by G = hA i. The restriction of A to any fixed block K ∈ F is an v,s v,s automorphism of L(K) ∼= L(F3) whose order divides 2. By Example 3.3, the 2 restriction is either of the unique conjugacy type of order 2 or the identity. So the number of fixed points in K is either 3 or 7. Let F3 ⊆F denote the set of all fixed blocks with 3 fixed points and F7 those having 7 fixed points. We double count the set X of all pairs ({P1,P2},B) where {P1,P2} is a point-orbit of length 2 and B is a fixed block of D passing through P1 and P2. By Lemma 3.2, the number of choices for {P1,P2} is 2v−s−1(2s −1). By the design property, there is exactly λ = 1 block B of D passing through the line L = hP1,P2iF2. Since {P1,P2} is fixed under the action of Av,s, so is L. This implies that every block B′ in the orbit of B passes through L, too. By λ=1, we get B′ =B, so B is a fixed block. Thus #X =2v−s−1(2s−1). Now we first count the possibilities for the fixed block B. Since B contains a point-orbit of length 2, necessarily B ∈ F3. By Example 3.3, each such B contains exactly two point-orbits of length 2. So #X = 2F3, which verifies equation (3). Now we double count the set Y of all pairs ({P1,P2},B) where P1,P2 are two distinct fixed points and B is a fixed block of D passing through P1 and P2. By Lemma 3.2, the number of choices for {P1,P2} is 2v−2s−1 = (2v−s − 1)(2v−s−1−1). As above, there is a single fixed block of D passing through P1 and P2, which yields #Y =(2v−s−1)(2v−s−1−1). (cid:0) (cid:1) 3 Vice versa,forB ∈F3 thereare 2 =3 choicesfor {P1,P2}andforB ∈F7 7 the number of choices is 2 = 21.(cid:0)S(cid:1)o #Y = 3F3 +21F7. This shows that (2v−s−1)(2v−s−1−1)= 3F3+21F7. Replacing F3 by formula (3), we obtain (cid:0) (cid:1) formula (4). Corollary 3. Let D be an S2[2,3,v] q-Steiner system. (a) In the case v ≡ 1mod6, D does not have an automorphism of type A v,s with 3∤s. (b) In the case v ≡ 3mod6, D does not have an automorphism of type A v,s with s≡2mod3. Proof. By definition, the number F7 of Lemma 3.5is aninteger. So the numer- ator of the right hand side of (4) must be a multiple of 7. 9 We compute its remainder modulo 7. The multiplicative order of 2 modulo 7 equals 3, so the remainder only depends on the values of v and s modulo 3. We get: s≡0mod3 s≡1mod3 s≡2mod3 v ≡1mod6 0 1 1 v ≡3mod6 0 0 −1 This concludes the proof. Now Theorem 2, which was stated in Section 1, follows as a combination of Corollary 3 and Lemma 3.1. 4. Automorphisms of a binary q-analog of the Fano plane This sectionis dedicatedtothe proofofTheorem1. Weprovethis assertion byeliminatingallothersubgroupsofGL(7,2). Thiseliminationisobtainedbya combinationof the theoreticalresults of Section 3 and a computer aided search based on the Kramer-Mesner method. Inthefollowing,weshalldenotebyDaputativeS2[2,3,7]q-Steinersystem. ItsautomorphismgroupAut(D)isasubgroupofthegroupGL(7,2)∼=PΓL(F7) 2 which has the order #GL(7,2)=221·34·51·72·311·1271. (5) According to the Sylow theorems, for each prime power pr with pr | #Aut(D) there exists a subgroup G ≤ GL(7,2) such that D is G-invariant. Hence for each prime factor p ∈ {2,3,5,7,31,127},we strive to find a (preferably small) exponent r such that there is no subgroup G ≤ GL(7,2) of order pr admitting a G-invariant S2[2,3,7] Steiner system. Then, we can conclude that Aut(D) is not divisible by pr. Since the automorphism groups of isomorphic q-Steiner systems are conjugate, it is sufficient to restrict the search to representatives of the subgroups of GL(7,2) up to conjugacy. Representatives of the conjugacy classes of the elements are given by the Jordan normal forms (with a fixed order of the Jor- danblocks). This providesanefficient wayto createthe cyclic subgroupsup to conjugacy.2 For general subgroup generation, the software package Magma is used [3]. For all the cases where subgroups were excluded computationally using the Kramer-Mesner method, details are given in Table 1. The column “group” contains the group in question. Its isomorphism type as an abstract group is described in column “type”. The columns “T-orb” and “K-orb” contain in- formation on the induced partition of the 2-subsets and 3-subsets, respectively. 2Weremarkthattwonon-conjugateelements maygenerateconjugate subgroups. 10